ABSTRACT

Suppose that u is the permutation in Example 3.1. Because there are three (equivalent) ways to express the 3-cycle, two ways to express (24) = (42), and because the three cycles can be written in any order, there are 3 x 2 x (3 !) = 36 different looking ways to express its disjoint cycle factorization. (Some examples are: (153)(24}(6), (315)(6)(24), (42)(531)(6}, and so on.) Apart from equivalence and the order in which the cycles are written, however, the disjoint cycle factorization of a permutation is unique. Moreover, it is customary when expressing permutations using disjoint cycle notation to suppress the cycles of length 1 (corresponding to fixed points). In particular, the permutation from Example 3.1 is typically written u = (153)(24). Dm:oonoN 3.4 The cyde type of u E S,. is the partition of n whose parts are the lengths of the cycles in its disjoint cycle factorization. Two permutations are said to have the same cyde structure if their cycle types are the same.